Mathematisches Institutskolloquium

Das Mathematische Institutskolloquium richtet sich an ein breites mathematisches Publikum (mit Bachelor Abschluss in Mathematik). Es soll die Diskussionen über die mathematischen Spezialisierungen der verschiedenen Arbeitsgruppen am Institut fördern. Außerdem sollen auch Studierende (Master-Studierende und fortgeschrittene Bachelor-Studierende) durch das Kolloquium die Gelegenheit erhalten sich über aktuelle Themen der Mathematik zu informieren.

Wintersemester 2021/2022

  • Prof. Constantinos Siettos (University of Naples Federico II, Italy)
    "Numerical Solution of Nonlinear PDEs and Stiff Problems of ODEs with Random Projection Networks and Extreme Learning Machines"
    Abstract: We use a class of machine learning the so-called Random Projections Networks and Extreme Learning Machines to numerically solve nonlinear partial differential equations (PDEs) and stiff problems of ODEs. For our demonstrations, we study several benchmark problems, namely (a) the Rober and Van-der Pol ODEs and, (b) the one-dimensional viscous Burgers and, the one- and two-dimensional Bratu PDEs. We also show how one can expolit the proposed methodology to construct bifurcation diagrams past limit points. The numerical efficiency of the proposed numerical machine learning scheme is compared against well established numerical analysis methods such as the adaptive Runge-Kutta ode45 and the ode15s a variable-step, variable-order solver based on the numerical differentiation formulas for solving the ODEs and central finite differences (FD) and Finite-element (FEM) methods for solving PDEs. We show that the proposed machine learning framework, regarding the solution of ODEs yields good numerical approximation accuracy without being affected by the stiffness, thus outperforming in same cases the ode45 and ode15s integrators, while regarding the solution of PDEs outperforms FD and importantly FEM for medium to large sized grids in terms of computational times and numerical accuracy.
    13.10.2021, 15:15 UhrOnline-Veranstaltung

Sommersemester 2021

  • Dr. Paolo Di Tella (University of Rostock)
    "On Martingale Representation Theorems"
    Abstract:  A central result in Stochastic Analysis is the martingale representation theorem of the Brownian motion. In this talk we are going to present classical result and extensions of the Brownian martingale representation theorem in more general contexts. We shall then present some applications to mathematical finance.
    14.07.2021, 15:15 Uhr, Online-Veranstaltung
  • Prof. Kim Knudsen (Technical University of Denmark)
    "Electromagnetic imaging – mathematical analysis and computations"
    In this talk we will look at inverse problems related to electromagnetic imaging. One example is Electrical Impedance Tomography (EIT), mathematically known as the Calderón problem, where the goal is to identify a body’s interior electrical conductivity distribution from measurements of voltages and currents on the surface of the body. This problem is severely ill-posed and requires heavy regularization techniques to be implemented before allowing for image reconstruction even with low resolution and contrast.
    Recently, novel hybrid imaging methods such as Acousto-Electric Tomography and Magnetic Resonance EIT have appeared. These approaches exploit different coupled physical phenomena and therefore hold promise for much more accurate and stable methods than EIT.
    From a mathematical analysis and computational perspective we consider the three different examples; we will formulate the relevant models, pose fundamental questions and give (partial) answers.
    16.06.2021, 15:15 Uhr, Online-Veranstaltung

Sommersemester 2020

  • Prof. Dr. Bernd Sturmfels (Direktor am Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig; Professor University of California at Berkeley)
    "3264 Conics in a Second"
    Abstract: Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given instance. This lecture illustrates how these two fields complement each other, especially in the light of emerging new applications. We start with a gem from19th century geometry, namely the 3264 conics that are tangent to five given conics in the plane. This topic was featured in the January 2020 issue of the Notices of the American Mathematical Society.  We conclude with an application in statistics, namely maximum likelihood estimation for linear Gaussian covariance models.
    17.06.2020, 15:00 Uhr, Online-Veranstaltung
  • Prof. Dr. Bernold Fiedler (FU Berlin)
    "Good to be late, precisely"
    13.05.2020, 15:15 - 16:15 Uhr, Online-Veranstaltung
  • Prof. Dr. Armin Iske (Universität Hamburg)
    "Kernel-based approximation methods for data analysis and machine learning"
    15.04.2020, 15:00 - 16:00 Uhr, Online-Veranstaltung

Wintersemester 2019/2020

  • Prof. Dr. Reinhard Racke (Universität Konstanz)
    "Exponentielle Stabilität für thermoelastische Systeme — die Multiplikatormethode"
    Abstract: Anhand der Wärmeleitungsgleichung bzw. der gedämpften Wellengleichung  wird die Energie- bzw. Multiplikatormethode bei Evolutionsgleichungen vorgestellt, ebenso ein Zusammenhang zur Fourierreihenentwicklung bzw. zur Fouriertransformation, letzteres am Beispiel thermoelastischer Platten. Numerische Resultate unterstreichen die analytisch erhaltenen Abschätzungen.
    13.11.2019, 15:15 Uhr, HS 326/327 (Ulmenstr. 69, Haus 3)
  • Prof. Dr. Markus Reiß (HU Berlin)
    "Statistics for stochastic PDEs"
    Abstract: For a broader audience we shall introduce to basic stochastic ordinary and partial differential equations (SODEs/SPDEs) and consider statistical estimation of the coefficients in these equations. We shall encounter a fundamental difference for drift estimation  for SODEs/SPDEs. Then we consider the specific problem of estimating the space-dependent diffusivity of a stochastic heat equation from time-continuous observations with space resolution h. This will be achieved by a localised Maximum-Likelihood approach. The rather counterintuitive convergence result and its efficiency as → 0 will be discussed in detail. 
    (joint work with Randolf Altmeyer, Berlin) 
    16.10.2019, 15:15 Uhr, HS 326/327  (Ulmenstraße 69, Haus 3)

Sommersemester 2019

  • Prof. Dr. Peter E. Kloeden (Universität Tübingen)
    "Random ordinary differential equations and their numerical approximation"
    Random ordinary differential equations (RODEs) are pathwise ordinary differential equations that contain a stochastic process in their vector field functions. They have been used for many years in a wide range of applications, but have been very much overshadowed by stochastic ordinary differential equations (SODEs). The stochastic process could be a fractional Brownian motion or a Poisson process, but when it is a diffusion process then there is a close connection between RODEs and SODEs through the Doss-Sussmann transformation and its generalisations, which relate a RODE and an SODE with the same (transformed) solutions. RODEs play an important role in the theory of random dynamical systems and random attractors.
    Classical numerical schemes such as Runge-Kutta schemes can be used for RODEs but do not achieve their usual high order since the vector field does not inherit enough smoothness in time from the driving process. It will be shown how, nevertheless, various kinds of Taylor-like expansions of the solutions of RODES can be obtained when the stochastic process has Hölder continuous or even measurable sample paths and then used to derive pathwise convergent numerical schemes of arbitrarily high order. The use of bounded noise and an application in biology will be considered.
    Xiaoying Han and P. E. Kloeden,
    Random Ordinary Differential Equations and their Numerical Solution,
    Springer Nature Singapore, 2017.
    10.07.2019, 15:15 Uhr, HS 326/327 (Ulmenstr. 69, Haus 3)
  • Prof. Dr. Petra Wittbold (Universität Duisburg-Essen)
    "Time-fractional stochastic conservation laws"
    19.06.2019, 13:00 Uhr, HS 326 (Ulmenstr. 69, Haus 3)
  • Prof. Dr. Frank Vallentin (Universität zu Köln)
    "Über das Färben geometrischer Graphen (Chromatic numbers of geometric graphs)"
    Abstract: A classical problem in discrete geometry (due to Hadwiger and Nelson) is to find the minimal number of colors one needs to color all points in the Euclidean plane so that no two points which are distance 1 apart receive the same color. Similar geometric coloring problems can be posed in the context of the hyperbolic plane, n-dimensional Euclidean spaces, n-dimensional spheres, or coloring the Voronoi tessellation of n-dimensional lattices. In this talk I will present a theoretical framework in which all these geometric coloring problems (assuming that the color classes are measurable sets) can be conveniently studied with the help of harmonic analysis and convex optimization.
    15.05.2019, 15:00 Uhr, HS 326/327 (Ulmenstr. 69, Haus 3)
  • Prof. Dr. Michael Dellnitz (University of Paderborn, Germany)
    "Glimpse of the Infinite – on the Approximation of the Dynamical
    Behavior for Delay and Partial Differential Equations"

    Abstract: Over the last years so-called set oriented numerical methods have been developed for the analysis of the long-term behavior of finite-dimensional dynamical systems. The underlying idea is to approximate the corresponding objects of interest – for instance global attractors or related invariant measures – by box coverings which are created via multilevel subdivision techniques. That is, these techniques rely on partitions of the (finite-dimensional) state space, and it is not obvious how to extend them to the situation where the underlying dynamical system is infinite-dimensional.
    In this talk we will present a novel numerical framework for the computation of finite-dimensional dynamical objects for infinite-dimensional dynamical systems. Within this framework we will extend the classical set oriented numerical schemes mentioned above to the infinite-dimensional context. The underlying idea is to utilize appropriate embedding techniques for the reconstruction of global attractors in a certain finite-dimensional space. We will also illustrate our approach by the computation of global attractors both for delay and for partial differential equations; e. g. the Mackey-Glass equation or the Kuramoto-Sivashinsky equation.
    10.04.2019, 17:00 Uhr, HS 326/327 (Ulmenstr. 69, Haus 3)

Wintersemester 2018/2019

  • Prof. John Sheekey (University College Dublin, Ireland)
    "Finite Fields and their (less famous) cousins"
    Abstract: Finite Fields (or Galois Fields) are algebraic structures with a finite number of elements, in which all the usual rules of addition, multiplication, and division for the real numbers hold. The classic example is the integers modulo a prime, and it is well known that a finite field consisting of q elements exists precisely when q is a prime power. Furthermore it is well known that removing the assumption of commutativity of multiplication does not yield any new structures; all finite division rings are in fact fields, the Wedderburn-Dickson Theorem.
    Less well-known are Finite Semifields; in a semifield, we do not assume associativity of multiplication. Non-trivial finite semifields do exist, as first shown by Dickson (1905), and there remain many open questions regarding their construction and classification. They have many interesting connections to topics such as Projective Planes, Spreads, Flocks, PN functions, and more. Recently new connections have been made with Coding Theory and Cryptography, which has lead to a renewed interest due to potential practical applications.
    In this talk we will introduce the topic of finite semifields, present some of the known classifications and constructions, outline their connections to other areas of interest, and mention the main open problems.
    16.01.2019, 15:15 Uhr, HS 326/327 (Ulmenstr. 69, Haus 3)
  • Prof. Dr. Christof Büskens (University of Bremen, Germany)
    "Entwicklung ist  teuer; - Mathematik unbezahlbar!
    Hochdimensionale nichtlineare Optimierung trifft industrielle Anforderungen"

    Abstract: Nichtlineare Optimierung ist zu einer Schlüsseltechnologie in modernen Anwendungen geworden. Ob bei der Bestimmung neuer Kennfelder im Diesel-Abgasskandal oder dem Entwurf modellprädiktiver Regelungsverfahren, das Interesse an Lösungsmethoden für hoch- und höchstdimensionale Optimierungsprobleme ist groß.
    WORHP (We Optimise Really Huge Problems) gehört zu einer neuen Generation von nichtlinearen Optimierungslösern, die sogenannte schwach-besetzte Strukturen ausnutzen. Im Gegensatz zu anderen etablierten und "gewachsen" NLP-Lösern wurde WORHP zunächst vollständig am Reißbrett entworfen und berücksichtigt aktuelle Architekturen, mathematische Entwicklungen, neue Computer- und Compilerstandards sowie industrielle Anforderungen. Hierdurch ist es möglich Probleme mit mehr als 1 Milliarde Variablen und Beschränkungen zu lösen. 
    Nach einer kurzen ingenieurwissenschaftlichen Motivation und einem Einblick in den aktuellen Stand zur nichtlinearen Optimierung, werden in diesem Vortrag Ideen aus der Echtzeitoptimierung ihren Weg in adaptive Regelungsverfahren finden. Abschliessend werden Anwendungen aus den Bereichen Automotive, Luft- und Raumfahrt sowie Energie vorgestellt. 
    12.12.2018, 15:00 Uhr, HS 326/327 (Ulmenstr. 69, Haus 3)